James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Appendix i Answers to Odd-Numbered Exercises A125
77. If c − 0, the graph is a cylindrical surface. For c . 0, the
level curves are ellipses. The graph curves upward as we leave
the origin, and the steepness increases as c increases. For c , 0,
the level curves are hyperbolas. The graph curves upward in the
y-direction and downward, approaching the xy-plane, in the
x-direction giving a saddle-shaped appearance near (0, 0, 1).
79. c − 22, 0, 2 81. (b) y − 0.75x 1 0.01
13.
20
z 0
_20
_2
x
0
2
_2
0
y
2
Exercises 14.2 • page 910
f sx, yd − x 2 y 3
1. Nothing; if f is continuous, f s3, 1d − 6 3. 2 5 2
5. 56 7. y2 9. Does not exist 11. Does not exist
13. 0 15. Does not exist 17. 2 19. s3
21. Does not exist
23. The graph shows that the function approaches different numbers
along different lines.
25. hsx, yd − s2x 1 3y 2 6d 2 1 s2x 1 3y 2 6;
2x 1 3y > 6j
hsx, yd |
27. Along the line y − x 29. R 2
31. hsx, yd | x 2 1 y 2 ± 1j 33. hsx, yd | x 2 1 y 2 < 1, x > 0j
35. hsx, y, zd | x 2 1 y 2 1 z 2 < 1j
37. hsx, yd | sx, yd ± s0, 0dj 39. 0 41. 21
43.
z
2
1
0
_1
_2
0
y
_2
0 x
2
Exercises 14.3 • page 923
2
f is continuous on R 2
1. (a) The rate of change of temperature as longitude varies, with
latitude and time fixed; the rate of change as only latitude varies;
the rate of change as only time varies
(b) Positive, negative, positive
3. (a) f Ts215, 30d < 1.3; for a temperature of 215°C and wind
speed of 30 kmyh, the wind-chill index rises by 1.3°C for each
degree the temperature increases. f vs215, 30d < 20.15; for a
temperature of 215°C and wind speed of 30 kmyh, the wind-chill
index decreases by 0.15°C for each kmyh the wind speed
increases.
(b) Positive, negative (c) 0
5. (a) Positive (b) Negative
7. (a) Positive (b) Negative
9. c − f, b − f x, a − f y
11. f xs1, 2d − 28 − slope of C 1, f ys1, 2d − 24 − slope of C 2
2
x
z
0
16
(1, 2)
(1, 2, 8)
C¡
4
y
x
2
z
0
16
C
(1, 2)
(1, 2, 8)
4
y
z
z
20
0
_20
_2
40
20
0
_2
x
0
2
_2
f xsx, yd − 2xy 3
0
x
2 _2
f ysx, yd − 3x 2 y 2
15. f xsx, yd − 4x 3 1 5y 3 , f ysx, yd − 15xy 2
17. f xsx, td − 2t 2 e 2x , f tsx, td − 2te 2x
−z
19.
−x − 1
x 1 t , −z
2 −t − 2t
x 1 t 2
21. f xsx, yd − 1yy, f ysx, yd − 2xyy 2
23. f xsx, yd −
sad 2 bcdy
sbc 2 addx
2
, fysx, yd −
scx 1 dyd scx 1 dyd 2
25. t usu, vd − 10uvsu 2 v 2 v 3 d 4 , t vsu, vd − 5su 2 2 3v 2 dsu 2 v 2 v 3 d 4
27. R psp, qd −
q 2
2pq
1 1 p 2 4
, Rqsp, qd −
q 1 1 p 2 q 4
29. F xsx, yd − cosse x d, F ysx, yd − 2cosse y d
31. f x − 3x 2 yz 2 , f y − x 3 z 2 1 2z, f z − 2x 3 yz 1 2y
33. −wy−x − 1ysx 1 2y 1 3zd, −wy−y − 2ysx 1 2y 1 3zd,
−wy−z − 3ysx 1 2y 1 3zd
35. −py−t − 2t 3 yst 4 1 u 2 cos v ,
−py−u − u cos vyst 4 1 u 2 cos v ,
−py−v − 2u 2 sin vy(2st 4 1 u 2 cos v )
37. h x − 2xy cosszytd, h y − x 2 cosszytd,
h z − s2x 2 yytd sinszytd, h t − sx 2 yzyt 2 d sinszytd
39. −uy−x i − x iysx
2
12 1 x 22 1 ∙∙∙ 1 x n
41. 1 43. 1 6 45. fxsx, yd − y 2 2 3x 2 y, f ysx, yd − 2xy 2 x 3
−z
47.
−x − 2 x
3z , −z
−y − 2 2y
3z
49.
−z
−x −
yz
e z 2 xy , −z
−y −
xz
e z 2 xy
0
0
y
y
2
2
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